Mathematics • Year 8 • Unit 4 • Lesson 19
Experimental Probability
Build fluency with calculating experimental probability (frequency ÷ trials), comparing it to theoretical, and applying the law of large numbers and expected frequencies.
1. I do — fully worked example
Watch how we calculate experimental probability, compare to theoretical, and find an expected frequency.
Problem. A die is rolled 60 times. The outcome 4 appears 13 times. (a) Find the experimental P(rolling a 4). (b) State the theoretical probability. (c) How many 4s would we expect in 600 rolls of a fair die?
Step 1 — Apply the experimental probability formula.
P(4) ≈ frequency / trials = 13 / 60
Reason: experimental probability is the fraction of trials where the event occurred.
Step 2 — Convert to decimal for easier comparison.
13 / 60 ≈ 0.217 (3 d.p.)
Step 3 — Find theoretical probability.
A fair die: P(4) = 1/6 ≈ 0.167
Reason: each face is equally likely, so 1 out of 6.
Step 4 — Compare and comment.
0.217 vs 0.167 → close, but 60 trials is a small sample. The difference (0.05) is within normal variation.
Step 5 — Expected frequency in 600 rolls.
Expected = P × n = (1/6) × 600 = 100
Answer: Experimental P ≈ 13/60 ≈ 0.217; theoretical P = 1/6 ≈ 0.167; expected 4s in 600 rolls = 100.
2. We do — fill in the missing steps
Same shape as Section 1, but with the working faded. Fill in each blank. 4 marks
Problem. A coin is flipped 80 times. Heads appears 36 times. (a) Find experimental P(H). (b) Compare to theoretical. (c) Expected heads in 800 flips of a fair coin.
Step 1 — Experimental formula:
P(H) ≈ ______ / ______
Step 2 — Decimal:
______ / ______ = ______ (3 d.p.)
Step 3 — Theoretical P(H):
P(H) = ______ / ______ = ______
Step 4 — Compare:
______ vs ______ → close / not close ? (circle)
Step 5 — Expected H in 800 flips:
Expected = ______ × 800 = ______
3. You do — independent practice
Show working under each problem. Foundation tests the formula, standard does the comparison, extension uses expected frequencies and the law of large numbers.
Foundation — recall and formula
3.1 A spinner is spun 40 times and lands on red 18 times. Calculate the experimental P(red). Simplify. 1 mark
3.2 Define theoretical probability in one sentence. Use the words "equally likely". 1 mark
3.3 State the law of large numbers in your own words. 1 mark
3.4 A die is rolled 600 times. State the expected frequency of rolling a 2. Show the calculation. 1 mark
Standard — calculate and compare
3.5 A die is rolled 120 times. The outcome 5 appears 18 times. Calculate the experimental P(rolling a 5) and compare to the theoretical probability. Is there strong evidence of bias? 2 marks
3.6 A 4-sided die (faces 1, 2, 3, 4) is rolled 200 times with this frequency table: 1→55, 2→48, 3→52, 4→45. Calculate the experimental probability for each face. Does the die appear fair? 2 marks
Extension — expected frequency and reasoning
3.7 A bag has 5 marbles (2 Red, 3 Blue). One marble is drawn 300 times with replacement. (a) State the theoretical P(Red). (b) State the expected frequency of Red. (c) If 130 Reds were observed, is this consistent with a fair bag? Explain. 2 marks
3.8 Two students each roll a die 30 times. Student A gets 7 sixes. Student B gets 3 sixes. (a) Calculate each student's experimental P(6). (b) If they combine their data (60 trials total), calculate the combined P(6). (c) Which estimate is most reliable and why? 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do (coin, 36 H in 80)
Step 1: P(H) ≈ 36 / 80. Step 2: 36/80 = 0.450. Step 3: Theoretical P(H) = 1/2 = 0.500. Step 4: close (within 0.05; 80 trials is a small sample so this variation is normal). Step 5: Expected = 0.5 × 800 = 400.
3.1 — Spinner, 18 reds in 40
P(red) ≈ 18 / 40 = 9/20 = 0.45.
3.2 — Theoretical probability
Theoretical probability is calculated by counting equally likely outcomes — favourable ÷ total — without running an experiment.
3.3 — Law of large numbers
As the number of trials increases, the experimental probability gets closer to the theoretical probability.
3.4 — Expected 2s in 600 rolls
Expected = (1/6) × 600 = 100.
3.5 — Die rolled 120 times, 18 fives
Experimental P(5) = 18/120 = 3/20 = 0.150. Theoretical P(5) = 1/6 ≈ 0.167. The difference is small (about 0.017), well within normal sampling variation for 120 trials. No strong evidence of bias.
3.6 — 4-sided die fairness
P(1) ≈ 55/200 = 0.275; P(2) ≈ 48/200 = 0.240; P(3) ≈ 52/200 = 0.260; P(4) ≈ 45/200 = 0.225. Theoretical = 0.250 for each. All four are close to 0.250 (within about 0.025). The die appears fair — the small variations are consistent with chance in 200 trials.
3.7 — Bag of marbles, 300 draws
(a) P(Red) = 2/5 = 0.4.
(b) Expected Red = 0.4 × 300 = 120.
(c) 130 is close to the expected 120 (10 above). This is well within normal variation for 300 trials — consistent with a fair bag.
3.8 — Two students, then combined
(a) A: P(6) = 7/30 ≈ 0.233. B: P(6) = 3/30 = 0.100.
(b) Combined: P(6) = (7+3)/60 = 10/60 = 1/6 ≈ 0.167.
(c) The combined estimate (60 trials) is more reliable — by the law of large numbers, more trials gives a value closer to the theoretical 1/6. Either individual estimate of 30 trials is noisy.